Once you know the slope, write the equation as $y=mx+b$ and plug in the slope and the coordinates of one point to solve for $b$.

After finding $m$ and $b$, use the other point to confirm the equation works, or compare your equation to the options to see which one matches.

In Desmos, type (-3,7) and (5,-9) on separate lines so both points appear on the graph.

On new lines in Desmos, type each option exactly as written, for example y=2x+1, y=-2x+1, y=-2x-1, and y=-1/2x+1. This will draw four different lines.

Look at which of the four lines passes through both plotted points $(-3,7)$ and $(5,-9)$. The equation of that line is the correct choice.

Use the slope formula for a line through $(x_1,y_1)$ and $(x_2,y_2)$:

Here, use $(-3,7)$ as $(x_1,y_1)$ and $(5,-9)$ as $(x_2,y_2)$:

So the slope of the line is $m=-2$.

Slope-intercept form is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept.

You already found $m=-2$, so write:

Now plug in the coordinates of one of the given points (for example, $(-3,7)$) to solve for $b$:

Subtract $6$ from both sides:

So the $y$-intercept is $b=1$.

Now substitute $m=-2$ and $b=1$ back into $y=mx+b$:

Since $y=g(x)$, the defining equation is $g(x)=-2x+1$, which is the correct choice.